Nonlinear approximation of functions by sets of finite pseudo-dimension in the probabilistic and average case settings

Abstract

This paper aims to explore the optimal nonlinear approximation on the class of functions in Sobolev space in the probabilistic and average case settings. Specifically, we investigate the optimal approximation through the utilization of a set with finite pseudo-dimension, measured by the Kolmogorov probabilistic nonlinear [Formula: see text]-width (or say, the Kolmogorov probabilistic pseudo-[Formula: see text]-width). Furthermore, we provide an estimation of the exact asymptotic order of the Kolmogorov probabilistic nonlinear [Formula: see text]-width and [Formula: see text]-average nonlinear [Formula: see text]-width on the class of functions in Sobolev space.